Keeping Time on the Moon: A Relativistic Approach to Lunar Clocks

Table of Links

Abstract and 1. Introduction

2. Clock in Orbit

   2.1 Coordinate Time

   2.2 Local Frame for the Moon

3. Clock Rate Differences Between Earth and Moon

4. Clocks at Earth-Moon Lagrance Points

   4.1 Clock at Lagrange point L1

   4.2. Clock at Lagrange point L2

   4.3. Clock at Lagrange point L4 or L5

5. Conclusions

Appendix 1: Fermi Coordinates with Origin at the Center of the Moon

Appendix 2: Construction of Freely Falling Center of Mass Frame

Appendix 3: Equations of Motion of Earth and Moon

Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems

Acknowledgments and References

APPENDIX 1: FERMI COORDINATES WITH ORIGIN AT THE CENTER OF THE MOON

We give the transformation equations between barycentric coordinates and Fermi normal coordinates with the center at the Moon as follows:[6]

Here, the notation (m) as in V(m) represents quantities evaluated at the Moon’s center of mass. The quantity V (m) is the magnitude of the Moon’s velocity. Transformation coefficients can be derived and are:

Transformation of the metric tensor is accomplished with the usual formula:

where the summation convention for repeated indices applies. Thus, for the time-time component of the metric tensor in the freely falling frame,

APPENDIX 2: CONSTRUCTION OF FREELY FALLING CENTER OF MASS FRAME

The transformation coefficients are easily obtained from the above coordinate transformations and are

Transformation of the metric tensor using Eq. (72): the metric component g00 in the center of mass frame,

Summarizing, the scalar invariant in the center of mass system is